The random variable \(T\) is the lifetime, in hours, of a randomly chosen battery of a particular type. It is given that \(T\) has a negative exponential distribution with mean 400 hours.
(i) Write down the probability density function of \(T\).

(ii) Find the probability that a battery of this type has a lifetime that is less than 500 hours.
(iii) Find the median of the distribution.